Statistical Modelling of Three-Dimensional Perturbation Growth in a Transitional Blasius Boundary Layer

Perturbations within a boundary layer can undergo significant, linear energy amplification, even when all eigenvalues of the linearized Navier-Stokes operator are stable. The behavior of a perturbation is often characterized by the energy growth of the optimal global mode or the optimal inlet condition. A statistical framework, developed by Frame & Towne (2024), demonstrates that the mean energy growth can deviate significantly from the optimal growth and furthermore that the evolution of a perturbation is more comprehensively described by the evolution of the energy PDF than by the energy of the optimal mode. We expand this framework to the spatial stability problem of a Mach 0.1 compressible Blasius boundary layer, demonstrating a large disparity between the downstream responses of the optimal inlet condition and an ensemble of realizations of inlet conditions drawn from a PDF with a physically informed correlation length. The downstream response is simulated using a linear One-Way Navier-Stokes solver and is used to estimate the mean and PDF of the perturbation energy amplification. The linear response across the physically relevant frequency-wave number space is analyzed to predict, for some transition threshold energy, the likelihood of laminar-to-turbulent transition as a function of the streamwise direction. Model predictions are compared to a companion direct numerical simulation.

Frame, P. & Towne, A. (2024). Beyond optimal disturbances: a statistical framework for transient growth. Journal of Fluid Mechanics, 983, A2.